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Title: Inference in the Fr\'echet regression model for object responses Authors:  Alexander Petersen - Brigham Young University (United States) [presenting]
Paromita Dubey - Stanford University (United States)
Hans-Georg Mueller - University of California Davis (United States)
Abstract: Linear regression is one of the most widely used fundamental tools in statistics for modeling response predictor relationships. In modern applications, one often encounters paired data where the responses are non-Euclidean objects (e.g., networks or covariance matrices) and predictors take values in $\mathbb{R}^d$. Fr\'echet regression is a state of the art method which is geared towards modeling regression between a metric space valued response and multivariate predictors. We propose the means to test the goodness of fit in Fr\'echet regression models, beginning with a test for the null hypothesis that the global Fr\'echet regression function does not depend on the predictors. We also extend the approach to investigate the partial effect of adding new predictors in an existing Fr\'echet regression model. The criteria we propose for the above tests are asymptotically degenerate under standard regularity conditions. To overcome this limitation, we propose the use of random multipliers, independent of the data, that give a non-degenerate distribution of the proposed test statistic under the null hypothesis of no regression effect. We illustrate the performance of the proposed test through multiple experiments in simulations, including samples of symmetric positive definite matrices equipped with various non-Euclidean metrics.